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Prove that the operation is not associative
Given:
\( a * b = \frac{a+b}{2}, \quad a,b \in \mathbb{Q} \)
Proof (Counterexample Method):
Take \( a = 1 \), \( b = 3 \), \( c = 5 \)
LHS:
\( (a*b)*c = \left(\frac{1+3}{2}\right)*5 = 2*5 \)
\( = \frac{2+5}{2} = \frac{7}{2} \)
RHS:
\( a*(b*c) = 1*\left(\frac{3+5}{2}\right) = 1*4 \)
\( = \frac{1+4}{2} = \frac{5}{2} \)
Clearly:
\( \frac{7}{2} \neq \frac{5}{2} \)
Conclusion:
❌ Therefore, the operation is NOT associative on \( \mathbb{Q} \).